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https://designers-guide.org/forum/YaBB.pl Design >> Analog Design >> stability for ckts having two feedback loops https://designers-guide.org/forum/YaBB.pl?num=1198131494 Message started by justdoit on Dec 19^{th}, 2007, 10:18pm |
Title: stability for ckts having two feedback loops Post by justdoit on Dec 19^{th}, 2007, 10:18pm Hi all , Can any one give me the references for doing the stability analysis if there are multiple feedback loops (i have two feedback loops in my circuit ) in circuit ... Thanks in advance |
Title: Re: stability for ckts having two feedback loops Post by buddypoor on Dec 20^{th}, 2007, 9:24am Hi Haribabu ! You have picked up a very interesting point in control theory since this question is not answered in most text books. I face this problem since several years and I came to the conclusion that there is no other way than to do an open loop analysis for each of the possible loops. In your case, if you have two feedback loops, there are three possible options: 1.)Loop 1 open, loop 2 closed 2.)Loop 2 open, loop 1 closed 3.) Both loops open. Than, the most critical stability margin determines the system margin. With other words, the loop with the least margin is dominant. Interestingly, to reduce the system to a single loop system prior to the open-loop-analysis gives exactly the same results. The reason is that - in your example - there are exactly three different ways to do this (with three different loop gains and, hence, three different margins). I hope this could help a little. Good luck and greetings from Germany. Lutz |
Title: Re: stability for ckts having two feedback loops Post by toseii on Dec 22^{nd}, 2007, 8:31pm Hi Haribabu, This topic is a very tough one. However the analysis of multi-loop feedback systems can be simplified to a single loop case if any of this two conditions is met: 1) All the feedback loops have a break point in common. In this case, by breaking the loops at that point the stability analysis is performed as if it were a single loop case. 2) A multi-loop feedback which comprises one general feedback network and several inner (local) feedback loops can be analyzed by just breaking the general feedback network (single loop) if each of its inner feedback networks is stable by itself (which must be left intact during the analisys). The paper "Determination of stability using return ratios in balanced fully differential feedback circuits" by Paul Hurst, IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS-11: ANALOG AND DIGITAL SIGNAL PROCESSING, VOL. 42, NO. 12, DECEMBER 1995 , has examples of the cases mentioned above Hope this helps tosei |
Title: Re: stability for ckts having two feedback loops Post by buddypoor on Dec 26^{th}, 2007, 12:32am The above reply from TOSEI is, of course, completely correct as it takes two special cases into consideration. The first example refers to the common case that an amplifier has one negative and one positive feedback circuit. The second example can be found in control circuits with one or more "local" loops and one overall feedback loop, which than is the most "slowest" of all. But, what is the procedure if none of these cases applies ? There are a lot of cases where, for example, two loops exist which cannot discriminated by "inner loop" resp. "outer loop". In this case, I think, the procedure has to be as described in my first reply to HARIBABU on Dec. 20th. Best wishes and a happy new year to all community members Lutz (Germany) |
Title: Re: stability for ckts having two feedback loops Post by thechopper on Jan 14^{th}, 2008, 10:56am Hi Lutz, From your previous reply, you suggested that buddypoor wrote on Dec 20^{th}, 2007, 9:24am:
My question is, how do you account for the influence of the two loops closed at the same time in your analysis?. I understand that by analyzing them in a separate way, the one with the least margin will dominate. However, things might be even worse when considering all the cases you enumerated at the same time, because of the relative margin between the different options. Could you please clarify? Thanks Tosei |
Title: Re: stability for ckts having two feedback loops Post by Eugene on Jan 14^{th}, 2008, 10:41pm I thought there was already a discussion of this somewhere in the Forum but I could not find it. Oh well, perhaps it's just a case of deja vu. Anyway, the most direct method of assessing stability of multiloop systems is to compute the Eigen values of the closed loop system matrix. This is not that practical in electric circuits because the poles are often widely separated and the order of the system can also be quite high. Spectre and SPICE have tools for computing closed loop poles but they sometimes list RHP (i.e. unstable) poles when none exist. Such poles are usually in close proxmity to closed loop zeros such that they nearly cancel. If you insist on using more classical frequency domain methods, there is a little known procedure called sequential loop closures, or sequential return differences. The procedure is as follows: 1. Open all loops such that the resulting system is stable. 2. Assess the loop gain of one loop with all other loops open. Keep track of the number of clockwise encirclements of the Nyquist point. 3. Close that loop and assess the loop gain of the next loop. Keep track of the net number of encirlcements of the Nyquist point. 4. Close that loop and do the same for the next loop. And so on. 5. If the net number of encirclements (clockwise - counter clockwise) equals zero, the system is stable. If the net is greater than zero, the system is unstable. The only problem with this method is that it is hard to identify a single phase margin or gain margin. You could select the minimum phase margin and minimum gain margin as you assessed each loop but you may get different numbers if you select a different sequence. Despite this shortcoming, the procedure is mathematically rigorous. This method is often mis-applied, most commonly when common mode feedback loops are involved. Suppose we have two interacting loops called loop one and loop two. The mistake is to start with loop one closed while you assess loop two. You then look at loop one with loop two closed. The problem is that the logic is circular because you do not know how many RHP poles you are starting with. If loop one has one RHP pole, loop two MUST encircle the Nyquist point exactly once counter clockwise to make the closed loop system stable. In short, the mis-applied method is the same as saying loop one is stable because loop two is stable and loop two is stable because loop one is stable, therefore the system is stable. Imagine two brothers going to court saying "I'm telling the truth because my brother never lies and he says I'm telling the truth". The other brother says the same thing. Does that prove they are both truthful? To get the net encirclements at the end of the procedure, you must start with all loops open; you must start knowing for sure that one brother never lies. |
Title: Re: stability for ckts having two feedback loops Post by Frank Wiedmann on Jan 14^{th}, 2008, 11:31pm Eugene wrote on Jan 14^{th}, 2008, 10:41pm:
You may have been thinking of http://www.designers-guide.org/Forum/YaBB.pl?num=1163532257/1#1 (see the last paragraph of reply #1). By the way, for Bode's method, you must open the loops not by cutting a wire but by setting the controlled sources to zero (so that the impedances you are seeing are not changed). This is very difficult with traditional circuit simulators because the controlled sources are usually inside transistor models and you do not have direct access to them. |
Title: Re: stability for ckts having two feedback loops Post by buddypoor on Jan 15^{th}, 2008, 2:45am Hello Haribabu! Q:My question is, how do you account for the influence of the two loops closed at the same time in your analysis? A: Stability margins (e.g. phase margin) can be defined only for an open loop. It is defined as the amount of additional (parasitic) phase shift which must be introduced into this loop in order to make the system instable if the loop is closed again. Therefore, when both loops of a two-loop system are closed you cannot perform any additional stability analysis (hopefully I did understand your question correctly). Eugene wrote on Jan 14^{th}, 2008, 10:41pm:
Of course, I completely agree with the procedure as quoted above. One short comment to the "problem" as mentioned above: If one system has three different feedback loops it certainly will exhibit three different stability margins. Therefore. to ask for a single system margin is - for my opinion - a more or less philosophical question, because in reality additional parasitic phase shifts will occur not only in one of these loops. I think the whole subject seems to be a very interesting one and I appreciate the discussion about it. Lutz |
Title: Re: stability for ckts having two feedback loops Post by buddypoor on Jan 15^{th}, 2008, 2:49am Hello Tosei, I apologize for the error I have made in context with your name (Haribabu was the originator of this topic). Lutz |
Title: Re: stability for ckts having two feedback loops Post by Eugene on Jan 15^{th}, 2008, 8:28am Frank, Thanks for the reference. However, I was thinking of another Forum thread. Perhaps I was originally thinking of some e-mail exchanges. You raises an excellent point regarding the application of classical frequency domain stability analysis to transistor circuits. It can indeed be hard to find convenient break points within each loop. But that is what Spectre's stability probe is for is it not? I think the problem arises in multiloop systems because Spectre's stability probe does not actually break the loop; I don't think you can use the probe to break the loops you have not yet assessed. |
Title: Re: stability for ckts having two feedback loops Post by rajdeep on Jan 15^{th}, 2008, 9:34am Hi all, I found this discussion fascinating!! Well, I am not a circuit designer but has grown tremendous interest in it for last couple of months or so. I mainly deal with verification of mixed signal circuits. Can anyone suggest me a good book where the issue of control theory and its application in circuit design have been discussed in detail?? I understand handling multi-loop systems has not been covered that much, but even the single-loop analysis would be okay for me to start with. I have the book by Allen and Hollberg, but somehow it does not provide me with enough FEEL of the thing (must be due to my shortcomings). I have started to look into Razavi and like it very much. But for the control part I would prefer to have a book that looks at the problem in a more generic way first and then shows its application in ckt design. Is there any book or good lecture report anything available? Too much asking... and sorry for any digression! Let the good talk continue.. Rajdeep IIT Kharagpur!! |
Title: Re: stability for ckts having two feedback loops Post by thechopper on Jan 15^{th}, 2008, 11:29am buddypoor wrote on Jan 15^{th}, 2008, 2:45am:
Lutz: I'm not completely sure about your statement concerning the validity of determining a single system margin. Actually my question was more or less pointing towards that very concept: I was not completely sure - from the method you suggested - you were accounting for the "cross-effects" between loops as for overall stability (or in other words for a single system's margin). Although you might have several loops in your system, I do not see why a single margin for the system could not be determined. From Eugene's answer it looks superposition is the answer (see below) Eugene: Thanks for pointing out the sequential loop closures. From your description I understand you are simply applying superposition as for the stability criteria; i.e. counting the net (cumulative) number of encirclements around the -1 point in the Nyquist plot resulting from each individual loop assessment looks to me like superposition. Am I interpreting it correctly? If so, then my first answer as for how to account the relative influence of each loop into the other ones is simply: superposition. This would not be crazy since, at the end of the day, we are analyzing linearly-modeled systems. Thanks Tosei |
Title: Re: stability for ckts having two feedback loops Post by Frank Wiedmann on Jan 15^{th}, 2008, 1:04pm Eugene wrote on Jan 15^{th}, 2008, 8:28am:
Spectre's stb analysis works differently but it can only handle single loops. The probe is simply a dummy element like a voltage source with 0V that you specify when you set up the stb analysis, so of course it cannot be used to break the loop. If anyone wants to dig really deep into this loop-gain stuff (single loop only), you can find on my webpage http://www.geocities.com/frank_wiedmann/loopgain.html some additional thoughts about the method used by the stb analysis (I call it "Tian's method" there) and a somewhat different method for analyzing loop gain called the "General Feedback Theorem" (GFT), which was developed by R. David Middlebrook. The associated discussion starting at http://groups.yahoo.com/group/Design-Oriented_Analysis_D-OA/message/40 might also be worth reading, even if some of the points are probably mainly of academic interest. |
Title: Re: stability for ckts having two feedback loops Post by Eugene on Jan 15^{th}, 2008, 10:15pm I don't think I would refer to sequential loop closures as a superposition method. I say that because I would apply superpostion to a very special kind of multiloop system. If all feedback loops have one node in common such that all loops could be broken at that single node, then you can indeed define a single loop gain and that loop gain is the SUPERPOSITION (i.e. sum) of all the individual loop gains. But in general, the sum of all loop gains is only part of the story. If you apply Mason's rule to find the overall loop gain, the second term in the expansion gives you the "superposition" case I described above. The third term is the sum of the products of all non-touching loops; the fourth term is the sum of the products of all three-non-touching loops; and so on. My point is that in general the loop gain is not a simple superposition of all the individual loop gains. Now that I think of it, I suppose you could use Mason's rule to define a single loop gain for a multiloop system and that loop gain could be used to define a unique phase margin and gain margin. It's just that Mason's rule can be tricky to apply and I don't know of any simulator that has a Mason's rule calculator. |
Title: Re: stability for ckts having two feedback loops Post by Frank Wiedmann on Jan 15^{th}, 2008, 11:59pm The difficulty with applying Mason's rule might be that usually not all paths in an electronic circuit are unidirectional (consider e.g. resistive feedback). The General Feedback Theorem can be used to transform a circuit into an equivalent diagram that contains only unidirectional paths (which may or may not correspond to actual components of the circuit), but it only applies for single loops. In my opinion, when looking at phase margin or gain margin, you should always examine how sensitive they are to variations of your circuit elements in order to see how far you are away from instability. The well-known relationship between the phase margin and the step response of a circuit was derived for a rather special single-loop configuration (see the links I gave above for details), I certainly would not expect it to be valid for complicated multi-loop configurations. |
Title: Re: stability for ckts having two feedback loops Post by buddypoor on Jan 16^{th}, 2008, 7:16am Eugene and Tosei: 1.) I doubt if any method to reduce the number of loops (signal flow graph or block diagram reduction) can solve the problem. The reason is as follows: If a system has three different loops you can define three different loop gains by introducing three different cut points. If you instead applies some rules to reduce the system having only one single loop there are exactly three different ways to do this - depending on the rules and their sequence you are using. As a result, you get again the same different loop gains exhibiting three different margins. (Of course, the overall transfer function does not depend on the kind of rules resp. their sequence). Finally, starting from the overall system transfer function H(s), there are - in the above example - exact three different ways to produce by division three different denumerators of the form "1+Fo", with Fo being the negative loop gain. 2.) Perhaps it makes sense to ask: Why are we interested in a phase or gain margin ? My answer is: Because of design uncertainties I would like to get a rough feeling about some additional parasitic phase shift resp. additional gain which is allowed anywhere in the system without coming to close to instability. Of course, for a single loop system I am able to ask for specific figures of the margin (having in mind the time response). However, if there are - for example - three loops with three different margins, does it makes sense to compute a "system margin" (like a "mean value" or something else) ? I think the only practical way is to find the most critical margin and to define the corresponding loop as dominant. In many cases this will be an "outer" loop to be identified by simple inspection. In this case, all inner local loops can remain closed and it is not important if they are accessible or not. Having identified this "critical" loop one can do something to enhance the margin if necessary. Lutz |
Title: Re: stability for ckts having two feedback loops Post by Eugene on Jan 16^{th}, 2008, 8:34am All, First let me say that I am thoroughly enjoying this discussion. Thank you everyone!! Frank is quite right about the problem when signal flows in two directions through part of a loop. I encountered that last year and I regret that I did not have time to really interpret the apparently inconsistent results I observed. When working with multiple loops I think it sometimes comes down to which loop you can easily access. It may be moot to worry about loops buried deep in some device for example. On the other hand, if there is a single, perhaps outermost loop that you can easily affect, then perhaps it makes sense to worry only about that one loop. I have seen a couple of analog optimization tools now that claim to compute closed loop poles and zeros of large transistor circuits. I think they are based on symbolic analysis instead of pole zero fits to frequency responses. Perhaps these new tools open the door to direct closed loop pole zero analysis of circuits so that we no longer have to use phase/gain margin. We could simply look for the pole closest to the imaginary axis. |
Title: Re: stability for ckts having two feedback loops Post by thechopper on Jan 16^{th}, 2008, 5:56pm Eugene, I agree with what you pointed out concerning superposition. It is true is only valid for a special cases of multi loop systems, but for the general case does not apply. Lutz, Concerning your item #1, I guess I'm still slightly confused. From what you are saying three different margins are possible depending on the sequence you use in your analysis. However - from a physical standpoint - the actual system (in its closed loop form) should have an actual margin which should be unique. At least that makes sense to me. So I do not find the physical meaning of having different possible margins (Obviously that margin cannot be found when the loop is closed, but the marginality for the system, either small or large, will exist). Am I interpreting something wrong? (possibly) As for your 2nd point, I agree that - for most practical cases - we must look for the dominant pole and perform the analysis disregarding the inner local loops by keeping them closed as you suggested. Actually that situation is a very common in circuits (which supports your criterion), where a simple opamp in a negative feedback configuration has a clear dominant loop (the "external feedback loop") and several inner local loops (GD capacitances of MOS devices for instance) which are negligible when it comes to the overall stability. However, this scenario is the one I pointed out at the very beginning of this discussion (see my first reply, case 2). So I guess the question is still open (at least for me, and from what I posed in the paragraph above) for those cases when there is no clear dominant loop... Tosei |
Title: Re: stability for ckts having two feedback loops Post by buddypoor on Jan 17^{th}, 2008, 12:54am Tosei, to overcome your confusion, go back to the definition of – let´s say – the phase margin. This is simply the amount of phase shift to be introduced into a loop (a physical existing loop !) in order to make the closed loop system unstable. If a system exhibits two loops this definition, of course, applies also to the second loop which normally has a different loop gain and, hence, another phase response than loop No. 1. Thus, according to the above definition, two different phase margins exist. To approach the problem from the other side – when you expect only one single system phase margin “phi” the question arises: Where (that means: into which physical existing loop) do you introduce this amount of “phi” to make the system oscillate ?? This question can´t be answered without knowing the loop this margins originates from. And, more than that, this can be verified by experiment in order to demonstrate the "physical meaning" of this statement. By the way, this exactly is the reason that every textbook on control theory contains the warning: You must not derive the loop gain expression from the transfer function by evaluating the denumerator “1+Fo” by allocating “Fo” to the negative loop gain. In a multi-loop system this procedure is ambiguous because several ways exist to produce this denumerator form. Finally, concerning the "dominant loop" another point of view: As commonly agreed by all contributors to this topic in a multi-loop system that loop will dominate (i.e. determine mostly the system behaviour) which is the outermost (principle of overall feedback). However, sometimes such a loop cannot be identified. To find an answer I have performed a lot of simulation runs for several systems and came to the (still preliminary) conclusion that in such cases that loop dominates which is the slowest one. This seems to be logical, does it not ? However, the problem is to determine the time constants of each loop. Hopefully I could express myself in a clear way as my knowledge of the english language is rather limited. Lutz |
Title: Re: stability for ckts having two feedback loops Post by thechopper on Jan 18^{th}, 2008, 7:12pm Hi Lutz, I thought carefully about your last post, and I think I'm with you now concerning the idea of a single margin does not make much sense if we associate one margin figure per loop and therefore we know where we can introduce the necessary phase shift in order to make the system oscillate. However now I´m inclined to think this will be valid only when the margin of each loop does NOT affect the margin of any other loop in the system and the margin of different loops are comparable. I´m not sure if this is a valid physical scenario (think it its though), but since Mason's gain rule involves cross products of loop gains, then I think margins of each loop could also affect margins of other loops. If this was valid, then would not make sense to think of a single margin for the system? (Also, for the dominant loop case you describe I think it also make sense to treat that loop´s margin as the only margin in the system. This would be another case where you could consider single margin) Sorry I´m so hard to convince :) Tosei |
Title: Re: stability for ckts having two feedback loops Post by buddypoor on Jan 19^{th}, 2008, 8:02am Hello Tosei, Quote: However now I´m inclined to think this will be valid only when the margin of each loop does NOT affect the margin of any other loop in the system and the margin of different loops are comparable. As far as I understood the discussion up to now we only speak about systems exhibiting more than one feedback loop which have at least one common node. Of course, two loops which are completely isolated (decoupled) do not influence each other. Quote: I´m not sure if this is a valid physical scenario (think it its though), but since Mason's gain rule involves cross products of loop gains, then I think margins of each loop could also affect margins of other loops. If this was valid, then would not make sense to think of a single margin for the system? Of course, such a margin could be defined (more or less artificially ?), however, what would be the practical value of such a definition ? To make the problem clear, I´ll try do describe a simple system which is a rather common in electronics: A composite amplifier consisting of two opamps and four resistors. EXAMPLE: The output A1 of opamp 1 is connected to the noninverting input of opamp 2 which has a local feedback loop with resistors R1, R2 producing a gain of 10. The output A2 of this amplifier is fed back via a resistive divider (R3:R4=1:99) to the inverting input of opamp 1 thus producing an overall positive gain of 100. There are two realistice loop gain alternatives: Case (1) Open the main loop at node A1 (internal loop of opamp 2 still closed) Case (2) Open both loops at node A2. (A third alternative opening only loop 2 with loop 1 still closed has no practical meaning). Using PSpice and 741 opamp macros here are the simulated phase margins (ac analyses): (1) case 1 : 44 deg and (2) case 2: 41 deg. Now, if an additional pole at 100 kHz is introduced into the model of opamp 1 the results are as follows: (1) case 1: 13 deg and (2) case 2: margin negative, which means “unstable”. However, a TRAN analysis indicates that the system is stable (although the step response shows a remarkable ringing). From this it is to be concluded that in the above example opening both loops at the same time (case 2) does NOT lead to a loop gain which gives a reliable information about the circuit behaviour. This seems to be in contradiction to some earlier statements within this topic (e.g. reply 2 and 13). I think the discussion is (a) fascinating and (b) at the same time rather confusing because it seems that no general rule has been found up to now to check resp. to define the stability properties of multi-loop systems in the frequency domain. Regards Lutz |
Title: Re: stability for ckts having two feedback loops Post by thechopper on Feb 14^{th}, 2008, 2:47am buddypoor wrote on Jan 19^{th}, 2008, 8:02am:
Hi Lutz, Please let me insist one more time. I tried to replicate your example and I built a test bench using ideal opamp models which I parametrized such that they had similar open loop characteristics to the 741. What I found (AC analysis) when adding the extra 100KHz pole to the first opamp is that for case (1) (loop broken at A1 node) the margin was little smaller than what you got but still positive, and for the case (2) (loop broken at node A2) a large negative margin, indicating apparently the loop is not stable, and therefore in agreement with your previous statement. However these conclusions were derived from the Bode plot of those loops, which I'm almost sure is NOT applicable for the second case. The reason for this being : 1) The phase shift characteristic (at least in my simulation) crosses the 180 point at least twice 2) If you derive the stability from the corresponding Nyquist plot the system indicates is stable (as opposed to the Bode plot) My TRAN simulation looked similar to what you described: heavy ringing but attenuated, indicating a very small phase margin. Conclusion: Previous discussions in this forum concluded the Nyquist stability criterion is the general one and is always applicable while the Bode method is not. Based on this (and on my results), statements on previous replies #2 and 13 are still valid. The result of this is that we still have valid results from both cases which would allow us to define a single margin (not sure how) for the multi-loop system, which is the primary idea we were discussing. I might be doing something wrong and I also do not know how you concluded case (2) was unstable, but I´m pretty sure Nyquist criterion is the one telling the truth here. Regards Lutz |
Title: Re: stability for ckts having two feedback loops Post by buddypoor on Feb 16^{th}, 2008, 12:59am Hi, Tosei ! I agree with you that my conclusion in my last reply concerning BODE-stability was not correct; the reason was simply that I have stopped the simulation to early - i.e. at a frequency which was too low to discover the second point with 180 deg phase shift. So you are right, of course, that in cases with a magnitude response without a monotonic decrease not Bode but Nyquist tells the truth about stability. However, this does not solve our basic problem. Since the Nyquist plot is based on the loop gain (and not the closed loop transfer function) there still remains the question as mentioned at the beginning of this topic for a multi-loop system: Which loop gives the correct information ? I think, in these cases there is no other choice than to use a general – and some more complicated – stability criterion which does not need the loop gain but is based on an evaluation of the characteristic system equation in the time domain (e. g. Routh, Hurwitz). Thank you, Regards Lutz |
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